center of the centralizer of semisimple element

Question

Let $G$ be an adjoint group over an algebraically closed field $k$ and $s\in G$ a semisimple element.

Let $H=C_{G}(s)^{0}$ the neutral component of the centralizer of $s$. Do we have that the center of $H$ is itself connected?

Answer 1

{EDIT] Since I was discussing a slightly different question than the one actually asked, I'll start over with your connected group $H$ and look at its center.

The question illustrates some of the fine points in the structure theory of reductive groups, which are worth considering beyond Mikhail's counterexample (which I guess has to require good characteristic, here different from 2). In the formulation of the question, I'm assuming that $G$ is meant to be connected as well as semisimple of adjoint type over an algebraically closed field. Here the general result for a semisimple element $s \in G$ is that the centralizer is reductive but not necessarily connected (unless $G$ itself is simply connected). Your group $H$ is both connected and reductive.

More precisely, it's a familiar result that the centralizer $C_G(s)$ of $s$ is generated by some connected subgroups of $G$ (including a maximal torus) along with perhaps part of the Weyl group. Moreover, those connected subgroups are enough to generate the identity component $H$, which contains all unipotent elements of the centralizer (a relevant issue in characteristic $p>0$).

I think the basic problem you encounter is that while $G$ itself is of adjoint type, there is no reason why the connected semisimple derived group of $H$ should also be of adjoint type. So the center of $H$ might well be the direct product of the nontrivial finite center of this derived group and a torus. Beyond Mikhail's proposed example, I expect that a transparent example might be seen when $G$ has type $G_2$: this group is both simply connected and of adjoint type, but will contain subgroups isomorphic to $\mathrm{SL}_2$ or $\mathrm{SL}_3$ that aren't of adjoint type.

P.S. To answer the original question more systematically, I should refer to the notion of pseudo-Levi subgroup defined by my colleague Eric Sommers IMRN 1998: see 2.1. (He and I disagree about applying this label as he does to actual Levi subgroups of parabolics.) This comes from the idea of Borel and de Siebenthal, in line with Allen's comment. It is well known that pseudo-Levi subgroups coincide with the connected centralizers of semisimple elements, as written down by Eric's thesis adviser George Lusztig in 5.5 of his paper IMRN 1995. For example, the simple group of type $G_2$ (which is both adjoint and simply connected) has such a subgroup isomorphic to $\mathrm{SL}_3$ (which visibly has a nontrivial center unless the characteristic is 3).